ISSN 1464-8997 (on line) 1464-8989 (printed)
239
Geometry & Topology Monographs
Volume 3: Invitation to higher local fields
Part II, section 4, pages 239–253
4. Drinfeld modules and
local fields of positive characteristic
Ernst–Ulrich Gekeler
The relationship between local fields and Drinfeld modules is twofold. Drinfeld modules allow explicit construction of abelian and nonabelian extensions with prescribed
properties of local and global fields of positive characteristic. On the other hand,
n -dimensional local fields arise in the construction of (the compactification of) moduli schemes X for Drinfeld modules, such schemes being provided with a natural
stratification X0 ⊂ X1 ⊂ · · · Xi · · · ⊂ Xn = X through smooth subvarieties Xi of
dimension i.
We will survey that correspondence, but refer to the literature for detailed proofs
(provided these exist so far). An important remark is in order: The contents of this
article take place in characteristic p > 0 , and are in fact locked up in the characteristic p
world. No lift to characteristic zero nor even to schemes over Z/p2 is known!
4.1. Drinfeld modules
Let L be a field of characteristic p containing the field Fq , and denote by τ = τq
raising to the q th power map x 7→ xq . If “ a ” denotes multiplication by a ∈ L , then
τ a = aq τ . The ring End(Ga/L ) of endomorphisms of the additive group Ga/L equals
P
L{τp } = { ai τpi : ai ∈ L} , the non-commutative polynomial ring in τp = (x 7→ xp )
with the above commutation rule τp a = ap τ . Similarly, the subring EndFq (Ga/L ) of
Fq -endomorphisms is L{τ } with τ = τpn if q = pn . Note that L{τ } is an Fq -algebra
since Fq ,→ L{τ } is central.
Definition 1. Let C be a smooth geometrically connected projective curve over Fq . Fix
a closed (but not necessarily Fq -rational) point ∞ of C . The ring A = Γ(C−{∞}, OC )
is called a Drinfeld ring. Note that A∗ = Fq∗ .
c Geometry & Topology Publications
Published 10 December 2000: 240
E.-U. Gekeler
Example 1. If C is the projective line P1 /Fq and ∞ is the usual point at infinity then
A = Fq [T ].
Example 2. Suppose that p 6= 2 , that C is given by an affine equation Y 2 = f (X)
with a separable polynomial f (X) of even positive degree with leading coefficient a
non-square in Fq , and that ∞ is the point above X = ∞. Then A = Fq [X, Y ] is a
Drinfeld ring with degFq (∞) = 2 .
Definition 2. An A -structure on a field L is a homomorphism of Fq -algebras (in
brief: an Fq -ring homomorphism) γ: A → L . Its A -characteristic charA (L) is the
maximal ideal ker(γ), if γ fails to be injective, and ∞ otherwise. A Drinfeld module
structure on such a field L is given by an Fq -ring homomorphism φ: A → L{τ } such
that ∂ ◦ φ = γ , where ∂: L{τ } → L is the L -homomorphism sending τ to 0.
Denote φ(a) by φa ∈ EndFq (Ga/L ); φa induces on the additive group over L
(and on each L -algebra M ) a new structure as an A -module:
(4.1.1)
a ∗ x := φa (x)
(a ∈ A, x ∈ M ).
We briefly call φ a Drinfeld module over L , usually omitting reference to A.
Definition 3. Let φ and ψ be Drinfeld modules over the A -field L . A homomorphism
u: φ → ψ is an element of L{τ } such that u ◦ φa = ψa ◦ u for all a ∈ A. Hence
an endomorphism of φ is an element of the centralizer of φ(A) in L{τ } , and u is an
isomorphism if u ∈ L∗ ,→ L{τ } is subject to u ◦ φa = ψa ◦ u.
Define deg: a → Z∪{−∞} and degτ : L{τ } → Z∪{−∞} by deg(a) = logq |A/a|
( a 6= 0 ; we write A/a for A/aA ), deg(0) = −∞, and degτ (f ) = the well defined
degree of f as a “polynomial” in τ . It is an easy exercise in Dedekind rings to prove
the following
Proposition 1. If φ is a Drinfeld module over L , there exists a non-negative integer r
such that degτ (φa ) = r deg(a) for all a ∈ A ; r is called the rank rk(φ) of φ.
Obviously, rk(φ) = 0 means that φ = γ , i.e., the A -module structure on Ga/L is
the tautological one.
Definition 4. Denote by Mr (1)(L) the set of isomorphism classes of Drinfeld modules
of rank r over L .
Example 3. Let A = Fq [T ] be as in Example 1 and let K = Fq (T ) be its fraction
field. Defining a Drinfeld module φ over K or an extension field L of K is equivalent
to specifying φT = T + g1 τ + · · · + gr τ r ∈ L{T } , where gr 6= 0 and r = rk(φ). In
the special case where φT = T + τ , φ is called the Carlitz module. Two such Drinfeld
modules φ and φ0 are isomorphic over the algebraic closure Lalg of L if and only if
i
∗
there is some u ∈ Lalg such that gi0 = uq −1 gi for all i > 1 . Hence Mr (1)(Lalg ) can
Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields
Part II. Section 4. Drinfeld modules and local fields of positive characteristic
241
be described (for r > 1 ) as an open dense subvariety of a weighted projective space of
dimension r − 1 over Lalg .
4.2. Division points
Definition 5. For a ∈ A and a Drinfeld module φ over L , write a φ for the subscheme
of a -division points of Ga /L endowed with its structure of an A -module. Thus for
any L -algebra M ,
a φ(M ) = {x ∈ M : φa (x) = 0}.
T
More generally, we put a φ =
φa for an arbitrary (not necessarily principal) ideal
a∈a
a of A. It is a finite flat group scheme of degree rk(φ) · deg(a), whose structure is
described in the next result.
Proposition 2 ([ Dr ], [ DH, I, Thm. 3.3 and Remark 3.4 ]). Let the Drinfeld module φ
over L have rank r > 1 .
(i) If charA (L) = ∞, a φ is reduced for each ideal a of A, and a φ(Lsep ) = a φ(Lalg )
is isomorphic with (A/a)r as an A -module.
(ii) If p = charA (L) is a maximal ideal, then there exists an integer h , the height ht(φ)
of φ, satisfying 1 6 h 6 r , and such that a φ(Lalg ) ' (A/a)r−h whenever a is a
power of p , and a φ(Lalg ) ' (A/a)r if (a, p) = 1 .
The absolute Galois group GL of L acts on a φ(Lsep ) through A -linear automorphisms. Therefore, any Drinfeld module gives rise to Galois representations on its
division points. These representations tend to be “as large as possible”.
The prototype of result is the following theorem, due to Carlitz and Hayes [ H1 ].
Theorem 1. Let A be the polynomial ring Fq [T ] with field of fractions K . Let
ρ: A → K{τ } be the Carlitz module, ρT = T + τ . For any non-constant monic
polynomial a ∈ A, let K(a) := K(a ρ(K alg )) be the field extension generated by the
a -division points.
(i) K(a)/K is abelian with group (A/a)∗ . If σb is the automorphism corresponding
to the residue class of b mod a and x ∈ a ρ(K alg ) then σb (x) = ρb (x).
(ii) If (a) = pt is primary with some prime ideal p then K(a)/K is completely
ramified at
Q p and unramified at the other finite primes.
(iii) If (a) =
ai ( 1 6 i 6 s ) with primary and mutually coprime ai , the fields
K(ai ) are mutually linearly disjoint and K = ⊗i6i6s K(ai ).
(iv) Let K+ (a) be the fixed field of Fq∗ ,→ (A/a)∗ . Then ∞ is completely split in
K+ (a)/K and completely ramified in K(a)/K+ (a).
(v) Let p be a prime ideal generated by the monic polynomial π ∈ A and coprime
with a. Under the identification Gal(K(a)/K) = (A/a)∗ , the Frobenius element
Frobp equals the residue class of π mod a.
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E.-U. Gekeler
Letting a → ∞ with respect to divisibility, we obtain the field K(∞) generated
over K by all the division points of ρ , with group Gal(K(∞)/K) = lim a (A/a)∗ ,
−→
which almost agrees with the group of finite idele classes of K . It turns out that K(∞)
is the maximal abelian extension of K that is tamely ramified at ∞, i.e., we get a
constructive version of the class field theory of K . Hence the theorem may be seen
both as a global variant of Lubin–Tate’s theory and as an analogue in characteristic p
of the Kronecker–Weber theorem on cyclotomic extensions of Q .
There are vast generalizations into two directions:
(a) abelian class field theory of arbitrary global function fields K = Frac(A), where
A is a Drinfeld ring.
(b) systems of nonabelian Galois representations derived from Drinfeld modules.
As to (a), the first problem is to find the proper analogue of the Carlitz module for
an arbitrary Drinfeld ring A. As will result e.g. from Theorem 2 (see also (4.3.4)),
the isomorphism classes of rank-one Drinfeld modules over the algebraic closure K alg
of K correspond bijectively to the (finite!) class group Pic(A) of A. Moreover,
these Drinfeld modules ρ(a) ( a ∈ Pic(A) ) may be defined with coefficients in the ring
OH+ of A -integers of a certain abelian extension H+ of K , and such that the leading
coefficients of all ρ(aa) are units of OH+ . Using these data along with the identification
of H+ in the dictionary of class field theory yields a generalization of Theorem 1 to the
case of arbitrary A. In particular, we again find an explicit construction of the class
fields of K (subject to a tameness condition at ∞ ). However, in view of class number
problems, the theory (due to D. Hayes [ H2 ], and superbly presented in [ Go2, Ch.VII ])
has more of the flavour of complex multiplication theory than of classical cyclotomic
theory.
Generalization (b) is as follows. Suppose that L is a finite extension of K = Frac(A),
where A is a general Drinfeld ring, and let the Drinfeld module φ over L have rank r .
For each power pt of a prime p of A, GL = Gal(Lsep /L) acts on pt φ ' (A/pt )r . We
thus get an action of GL on the p -adic Tate module Tp (φ) ' (Ap )r of φ (see [ DH, I
sect. 4 ]. Here of course Ap = lim A/pt is the p -adic completion of A with field of
←−
fractions Kp . Let on the other hand End(φ) be the endomorphism ring of φ, which
also acts on Tp (φ). It is straightforward to show that (i) End(φ) acts faithfully and (ii)
the two actions commute. In other words, we get an inclusion
(4.2.1)
i: End(φ) ⊗A Ap ,→ EndGL (Tp (φ))
of finitely generated free Ap -modules. The plain analogue of the classical Tate conjecture for abelian varieties, proved 1983 by Faltings, suggests that i is in fact bijective. This has been shown by Taguchi [ Tag ] and Tamagawa. Taking End(Tp (φ)) '
Mat(r, Ap ) and the known structure of subalgebras of matrix algebras over a field into
account, this means that the subalgebra
Kp [GL ] ,→ End(Tp (φ) ⊗Ap Kp ) ' Mat(r, Kp )
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Part II. Section 4. Drinfeld modules and local fields of positive characteristic
243
generated by the Galois operators is as large as possible. A much stronger statement is
obtained by R. Pink [ P1, Thm. 0.2 ], who shows that the image of GL in Aut(Tp (φ))
has finite index in the centralizer group of End(φ) ⊗ Ap . Hence if e.g. φ has no
“complex multiplications” over Lalg (i.e., EndLalg (φ) = A; this is the generic case
for a Drinfeld module in characteristic ∞ ), then the image of GL has finite index in
Aut(Tp (φ)) ' GL(r, Ap ). This is quite satisfactory, on the one hand, since we may
use the Drinfeld module φ to construct large nonabelian Galois extensions of L with
prescribed ramification properties. On the other hand, the important (and difficult)
problem of estimating the index in question remains.
4.3. Weierstrass theory
Let A be a Drinfeld ring with field of fractions K , whose completion at ∞ is denoted
by K∞ . We normalize the corresponding absolute value | | = | |∞ as |a| = |A/a| for
0 6= a ∈ A and let C∞ be the completed algebraic closure of K∞ , i.e., the completion
alg
alg
of the algebraic closure K∞ with respect to the unique extension of | | to K∞ .
By Krasner’s theorem, C∞ is again algebraically closed ([ BGS, p. 146 ], where also
other facts on function theory in C∞ may be found). It is customary to indicate the
strong analogies between A, K, K∞ , C∞ , . . . and Z, Q, R, C, . . . , e.g. A is a discrete
and cocompact subring of K∞ . But note that C∞ fails to be locally compact since
|C∞ : K∞ | = ∞.
Definition 6. A lattice of rank r (an r -lattice in brief) in C∞ is a finitely generated
(hence projective) discrete A -submodule Λ of C∞ of projective rank r , where the
discreteness means that Λ has finite intersection with each ball in C∞ . The lattice
function eΛ : C∞ → C∞ of Λ is defined as the product
Y
(4.3.1)
eΛ (z) = z
(1 − z/λ).
06=λ∈Λ
It is entire (defined through an everywhere convergent power series), Λ -periodic and
Fq -linear. For a non-zero a ∈ A consider the diagram
(4.3.2)
0 −−−−→ Λ −−−−→


ay
e
C∞ −−−Λ−→ C∞ −−−−→ 0




ay
φΛ
ay
e
0 −−−−→ Λ −−−−→ C∞ −−−Λ−→ C∞ −−−−→ 0
with exact lines, where the left and middle arrows are multiplications by a and φΛ
a is
defined through commutativity. It is easy to verify that
(i) φΛ
a ∈ C∞ {τ } ,
(ii) degτ (φΛ
a ) = r · deg(a) ,
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E.-U. Gekeler
Λ
(iii) a 7→ φΛ
a is a ring homomorphism φ : A → C∞ {τ } , in fact, a Drinfeld module
of rank r . Moreover, all the Drinfeld modules over C∞ are so obtained.
Theorem 2 (Drinfeld [ Dr, Prop. 3.1 ]).
(i) Each rank- r Drinfeld module φ over C∞ comes via Λ 7→ φΛ from some r -lattice
Λ in C∞ .
0
(ii) Two Drinfeld modules φΛ , φΛ are isomorphic if and only if there exists 0 6= c ∈
C∞ such that Λ0 = c · Λ .
We may thus describe Mr (1)(C∞ ) (see Definition 4) as the space of r -lattices
modulo similarities, i.e., as some generalized upper half-plane modulo the action of an
arithmetic group. Let us make this more precise.
r−1
Definition 7. For r >
S 1 let P (C∞ ) be the C∞ -points of projective r−1 -space and
r
r−1
Ω := P (C∞ ) − H(C∞ ), where H runs through the K∞ -rational hyperplanes
ωr ) belongs to Drinfeld’s half-plane Ωr if and only
of Pr−1 . That is, ω = (ω1 : . . . :P
if there is no non-trivial relation
ai ωi = 0 with coefficients ai ∈ K∞ .
Both point sets Pr−1 (C∞ ) and Ωr carry structures of analytic spaces over C∞
(even over K∞ ), and so we can speak of holomorphic functions on Ωr . We will not
give the details (see for example [ GPRV, in particular lecture 6 ]); suffice it to say that
locally uniform limits of rational functions (e.g. Eisenstein series, see below) will be
holomorphic.
Suppose for the moment that the class number h(A) = |Pic(A)| of A equals
one, P
i.e., A is a principal ideal domain. Then each r -lattice Λ in C∞ is free,
Λ = 16i6r Aωi , and the discreteness of Λ is equivalent with ω := (ω1 : . . . : ωr )
belonging to Ωr ,→ Pr−1 (C∞ ). Further, two points ω and ω 0 describe similar lattices
(and therefore isomorphic Drinfeld modules) if and only if they are conjugate under
Γ := GL(r, A), which acts on Pr−1 (C∞ ) and its subspace Ωr . Therefore, we get a
canonical bijection
e Mr (1)(C∞ )
Γ \ Ωr →
(4.3.3)
from the quotient space Γ \ Ωr to the set of isomorphism classes Mr (1)(C∞ ).
In the general case of arbitrary h(A) ∈ N , we let Γi := GL(Yi ) ,→ GL(r, k), where
Yi ,→ K r ( 1 6 i 6 h(A) ) runs through representatives of the h(A) isomorphism
classes of projective A -modules of rank r . In a similar fashion (see e.g. [ G1, II
sect.1 ], [ G3 ]), we get a bijection
(4.3.4)
·
[
16i6h(A)
e Mr (1)(C∞ ),
Γi \ Ωr →
which can be made independent of choices if we use the canonical adelic description of
the Yi .
Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields
Part II. Section 4. Drinfeld modules and local fields of positive characteristic
245
Example 4. If r = 2 then Ω = S
Ω2 = P1 (C∞ ) − −P1 (K∞ ) = C∞ − K∞ , which
+
half-planes) than
rather corresponds to C − R = H
H − (upper and lower
complex
ab
az +b
+
to H alone. The group Γ := GL(2, A) acts via c d (z) = cz +d , and thus gives
rise to Drinfeld modular forms on Ω (see [ G1 ]). Suppose moreover that A = Fq [T ]
as in Examples 1 and 3. We define ad hoc a modular form of weight k for Γ as a
holomorphic function f : Ω → C∞ that satisfies
+b
(i) f az
= (cz + d)k f (z) for ac db ∈ Γ and
cz +d
(ii) f (z) is bounded on the subspace {z ∈ Ω : inf x∈K∞ |z − x| > 1} of Ω .
Further, we put Mk for the C∞ -vector space of modular forms of weight k . (In the
special case under consideration, (ii) is equivalent to the usual “holomorphy at cusps”
condition. For more general groups Γ , e.g. congruence subgroups of GL(2, A),
general Drinfeld rings A, and higher ranks r > 2 , condition (ii) is considerably more
costly to state, see [ G1 ].) Let
(4.3.5)
X
Ek (z) :=
(0,0)6=(a,b)∈A×A
1
(az + b)k
be the Eisenstein series of weight k . Due to the non-archimedean situation, the sum
converges for k > 1 and yields a modular form 0 6= Ek ∈ Mk if k ≡ 0 (q − 1).
Moreover, the various Mk are linearly independent and
M
(4.3.6)
M (Γ) :=
Mk = C∞ [Eq−1 , Eq2 −1 ]
k>0
is a polynomial ring in the two algebraically independent Eisenstein series of weights
q−1 and q 2 −1 . There is an a priori different method of constructing modular forms via
Drinfeld modules. With each z ∈ Ω , associate the 2-lattice Λz := Az + A ,→ C∞ and
the Drinfeld module φ(z ) = φ(Λz ) . Writing φ(Tz ) = T + g(z)τ + ∆(z)τ 2 , the coefficients
g and ∆ become functions in z , in fact, modular forms of respective weights q − 1
and q 2 − 1 . We have ([ Go1 ], [ G1, II 2.10 ])
(4.3.7)
2
2
q+1
g = (T g − T )Eq−1 , : ∆ = (T q − T )Eq2 −1 + (T q − T q )Eq−
1.
The crucial fact is that ∆(z) 6= 0 for z ∈ Ω , but ∆ vanishes “at infinity”. Letting
j(z) := g(z)q+1 /∆(z) (which is a function on Ω invariant under Γ ), the considerations
of Example 3 show that j is a complete invariant for Drinfeld modules of rank two.
Therefore, the composite map
(4.3.8)
e M2 (1)(C∞ ) →
e C∞
j: Γ \ Ω →
is bijective, in fact, biholomorphic.
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E.-U. Gekeler
4.4. Moduli schemes
We want to give a similar description of Mr (1)(C∞ ) for r > 2 and arbitrary A, that
is, to convert (4.3.4) into an isomorphism of analytic spaces. One proceeds as follows
(see [ Dr ], [ DH ], [ G3 ]):
(a) Generalize the notion of “Drinfeld A -module over an A -field L ” to “Drinfeld
A -module over an A -scheme S → Spec A ”. This is quite straightforward. Intuitively,
a Drinfeld module over S is a continuously varying family of Drinfeld modules over
the residue fields of S .
(b) Consider the functor on A -schemes:
isomorphism classes of rank-r
r
M : S 7−→
.
Drinfeld modules over S
The naive initial question is to represent this functor by an S -scheme M r (1). This is
impossible in view of the existence of automorphisms of Drinfeld modules even over
algebraically closed A -fields.
(c) As a remedy, introduce rigidifying level structures on Drinfeld modules. Fix some
ideal 0 6= n of A. An n -level structure on the Drinfeld module φ over the A -field L
whose A -characteristic doesn’t divide n is the choice of an isomorphism of A -modules
e n φ(L)
α: (A/n)r →
(compare Proposition 2). Appropriate modifications apply to the cases where charA (L)
divides n and where the definition field L is replaced by an A -scheme S . Let Mr (n)
be the functor




 isomorphism classes of rank-r

r
M (n): S 7−→ Drinfeld modules over S endowed .




with an n-level structure
Theorem 3 (Drinfeld [ Dr, Cor. to Prop. 5.4 ]). Suppose that n is divisible by at least
two different prime ideals. Then Mr (n) is representable by a smooth affine A -scheme
M r (n) of relative dimension r − 1 .
In other words, the scheme M r (n) carries a “tautological” Drinfeld module φ of
rank r endowed with a level- n structure such that pull-back induces for each A -scheme
S a bijection
(4.4.1)
e Mr (n)(S),
M r (n)(S) = {morphisms (S, M r (n))} →
f 7−→ f ∗ (φ).
M r (n) is called the (fine) moduli scheme for the moduli problem Mr (n). Now the finite
group G(n) := GL(r, A/n) acts on Mr (n) by permutations of the level structures. By
functoriality, it also acts on M r (n). We let M r (1) be the quotient of M r (n) by G(n)
(which does not depend on the choice of n ). It has the property that at least its L -valued
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Part II. Section 4. Drinfeld modules and local fields of positive characteristic
247
points for algebraically closed A -fields L correspond bijectively and functorially to
Mr (1)(L). It is therefore called a coarse moduli scheme for Mr (1). Combining the
above with (4.3.4) yields a bijection
(4.4.2)
·
[
16i6h(A)
e M r (1)(C∞ ),
Γi \ Ωr →
which even is an isomorphism of the underlying analytic spaces [ Dr, Prop. 6.6 ]. The
most simple special case is the one dealt with in Example 4, where M 2 (1) = A1 /A , the
affine line over A.
4.5. Compactification
It is a fundamental question to construct and study a “compactification” of the affine
A -scheme M r (n), relevant for example for the Langlands conjectures over K , the
cohomology of arithmetic subgroups of GL(r, A), or the K -theory of A and K .
This means that we are seeking a proper A -scheme M r (n) with an A -embedding
M r (n) ,→ M r (n) as an open dense subscheme, and which behaves functorially with
respect to the forgetful morphisms M r (n) → M r (m) if m is a divisor of n . For
many purposes it suffices to solve the apparently easier problem of constructing similar
compactifications of the generic fiber M r (n) ×A K or even of M r (n) ×A C∞ . Note
that varieties over C∞ may be studied by analytic means, using the GAGA principle.
There are presently three approaches towards the problem of compactification:
(a) a (sketchy) construction of the present author [ G2 ] of a compactification M Γ of
MΓ , the C∞ -variety corresponding to an arithmetic subgroup Γ of GL(r, A) (see
(4.3.4) and (4.4.2)). We will return to this below;
(b) an analytic compactification similar to (a), restricted to the case of a polynomial ring
A = Fq [T ], but with the advantage of presenting complete proofs, by M. M. Kapranov
[ K ];
(c) R. Pink’s idea of a modular compactification of M r (n) over A through a generalization of the underlying moduli problem [ P2 ].
Approaches (a) and (b) agree essentially in their common domain, up to notation
and some other choices. Let us briefly describe how one proceeds in (a). Since there is
nothing to show for r = 1 , we suppose that r > 2 .
We let A be any Drinfeld ring. If Γ is a subgroup of GL(r, K) commensurable
with GL(r, A) (we call such Γ arithmetic subgroups), the point set Γ \ Ω is the
set of C∞ -points of an affine variety MΓ over C∞ , as results from the discussion of
subsection 4.4. If Γ is the congruence subgroup Γ(n) = {γ ∈ GL(r, A): γ ≡ 1 mod n} ,
then MΓ is one of the irreducible components of M r (n) ×A C∞ .
Definition 8. For ω = (ω1 , : . . . : ωr ) ∈ Pr−1 (C∞ ) put
r(ω) := dimK (Kω1 + · · · + Kωr )
and r∞ (ω) := dimK∞ (K∞ ω1 + · · · + K∞ ωr ).
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E.-U. Gekeler
Then 1 6 r∞ (ω) 6 r(ω) 6 r and Ωr = {ω | r∞ (ω) = r} . More generally, for
1 6 i 6 r let
Ωr,i := {ω: r∞ (ω) = r(ω) = i}.
·
S
Then Ωr,i = ΩV , where V runs through the K -subspaces of dimension i of K r and
r
ΩV is constructed from V in a similar way as is Ωr = ΩK r from C∞
= (K r ) ⊗ C∞ .
That is, ΩV = {ω ∈ P(V ⊗ C∞ ) ,→ Pr−1 (C∞ ): r∞ (ω) = r(ω) = i} , which has a
natural structure as analytic space of dimension dim(V ) − 1 isomorphic with Ωdim(V ) .
·
S
Finally, we let Ωr := {ω: r∞ (ω) = r(ω)} = 16i6r Ωr,i .
Ωr along with its stratification through the Ωr,i is stable under GL(r, K), so this
also holds for the arithmetic group Γ in question. The quotient Γ \ Ωr turns out to be
the C∞ -points of the wanted compactification M Γ .
Definition 9. Let Pi ,→ G := GL(r) be the maximal parabolic subgroup of matrices
with first i columns being zero. Let Hi be the obvious factor group isomorphic
GL(r − i). Then Pi (K) acts via Hi (K) on K r−i and thus on Ωr−i . From
e {subspaces V of dimension r − i of K r }
G(K)/Pi (K) →
we get bijections
(4.5.1)
e Ωr,r−i ,
G(K) ×Pi (K ) Ωr−i →
(g, ωi+1 : . . . : ωr ) 7−→ (0 : · · · : 0 : ωi+1 : . . . : ωr )g −1
and
(4.5.2)
e
Γ \ Ωr,r−i →
·
[
g∈Γ\G(K )/Pi (K )
Γ(i, g) \ Ωr−i ,
where Γ(i, g) := Pi ∩ g −1 Γg , and the double quotient Γ \ G(K)/Pi (K) is finite by
elementary lattice theory. Note that the image of Γ(i, g) in Hi (K) (the group that
effectively acts on Ωr−i ) is again an arithmetic subgroup of Hi (K) = GL(r − i, K),
and so the right hand side of (4.5.2) is the disjoint union of analytic spaces of the same
0
type Γ0 \ Ωr .
Example 5. Let Γ = Γ(1) = GL(r, A) and i = 1 . Then Γ \ G(K)/P1 (K) equals
the set of isomorphism classes of projective A -modules of rank r − 1 , which in turn
(through the determinant map) is in one-to-one correspondence with the class group
Pic(A).
Let FV be the image of ΩV in Γ \ Ωr . The different analytic spaces FV ,
corresponding to locally closed subvarieties of M Γ , are glued together in such a way
that FU lies in the Zariski closure F V of FV if and only if U is Γ -conjugate to a
K -subspace of V . Taking into account that FV ' Γ0 \ Ωdim(V ) = MΓ0 (C∞ ) for some
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Part II. Section 4. Drinfeld modules and local fields of positive characteristic
249
arithmetic subgroup Γ0 of GL(dim(V ), K), F V corresponds to the compactification
M Γ0 of MΓ0 .
The details of the gluing procedure are quite technical and complicated and cannot
be presented here (see [ G2 ] and [ K ] for some special cases). Suffice it to say that
for each boundary component FV of codimension one, a vertical coordinate tV may
be specified such that FV is locally given by tV = 0 . The result (we refrain from
stating a “theorem” since proofs of the assertions below in full strength and generality
are published neither in [ G2 ] nor in [ K ]) will be a normal projective C∞ -variety M Γ
provided with an open dense embedding i: MΓ ,→ M Γ with the following properties:
• M Γ (C∞ ) = Γ \ Ωr , and the inclusion Γ \ Ωr ,→ Γ \ Ωr corresponds to i;
• M Γ is defined over the same finite abelian extension of K as is MΓ ;
• for Γ0 ,→ Γ , the natural map MΓ0 → MΓ extends to M Γ0 → M Γ ;
• the FV correspond to locally closed subvarieties, and F V = ∪FU , where U runs
through the K -subspaces of V contained up to the action of Γ in V ;
• M Γ is “virtually non-singular”, i.e., Γ contains a subgroup Γ0 of finite index such
that M Γ0 is non-singular; in that case, the boundary components of codimension
one present normal crossings.
S
Now suppose that M Γ is non-singular and that x ∈ M Γ (C∞ ) = 16i6r Ωr,i
belongs to Ωr,1 . Then we can find a sequence {x} = X0 ⊂ · · · Xi · · · ⊂ Xr−1 = M Γ
of smooth subvarieties Xi = F Vi of dimension i. Any holomorphic function around
x (or more generally, any modular form for Γ ) may thus be expanded as a series in
tV with coefficients in the function field of F Vr−1 , etc. Hence M Γ (or rather its
completion at the Xi ) may be described through (r − 1) -dimensional local fields with
residue field C∞ . The expansion of some standard modular forms can be explicitly
calculated, see [ G1, VI ] for the case of r = 2 . In the last section we shall present at
least the vanishing orders of some of these forms.
Example 6. Let A be the polynomial ring Fq [T ] and Γ = GL(r, A). As results from
Example 3, (4.3.3) and (4.4.2),
r
∗
MΓ (C∞ ) = M r (1)(C∞ ) = {(g1 , . . . , gr ) ∈ C∞
: gr 6= 0}/C∞
,
∗
where C∞
acts diagonally through c(g1 , . . . , gr ) = (. . . , cq −1 gi , . . . ), which is the
open subspace of weighted projective space Pr−1 (q−1, . . . , q r −1) with non-vanishing
last coordinate. The construction yields
i
M Γ (C∞ ) = Pr−1 (q − 1, . . . , q r − 1)(C∞ ) =
·
[
16i6r
M i (1)(C∞ ).
Its singularities are rather mild and may be removed upon replacing Γ by a congruence
subgroup.
Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields
250
E.-U. Gekeler
4.6. Vanishing orders of modular forms
In this final section we state some results about the vanishing orders of certain modular
forms along the boundary divisors of M Γ , in the case where Γ is either Γ(1) = GL(r, A)
or a full congruence subgroup Γ(n) of Γ(1). These are relevant for the determination of
K - and Chow groups, and for standard conjectures about the arithmetic interpretation
of partial zeta values.
In what follows, we suppose that r > 2 , and put zi := ωωri ( 1 6 i 6 r ) for the
coordinates (ω1 : . . . : ωr ) of ω ∈ Ωr . Quite generally, a = (a1 , . . . , ar ) denotes a
vector with r components.
Definition 10. The Eisenstein series Ek of weight k on Ωr is defined as
X
1
Ek (ω) :=
.
(a1 z1 + · · · + ar zr )k
r
06=a∈A
Similarly, we define for u ∈ n−1 × · · · × n−1 ⊂ K r
X
1
Ek,u (ω) =
.
(a1 z1 + · · · + ar zr )k
06=a∈K r
a≡u mod Ar
These are modular forms for Γ(1) and Γ(n), respectively, that is, they are holomorphic,
satisfy the obvious transformation values under Γ(1) (resp. Γ(n) ), and extend to
sections of a line bundle on M Γ . As in Example 4, there is a second type of modular
forms coming directly from Drinfeld modules.
Definition 11. For ω ∈ Ωr write Λω = Az1 + · · · + Azr and eω , φω for the lattice
function and Drinfeld module associated with Λω , respectively. If a ∈ A has degree
d = deg(a),
X
ω
φa = a +
`i (a, ω)τ i .
16i6r·d
The `i (a, ω) are modular forms of weight q i − 1 for Γ . This holds in particular for
∆a (ω) := `rd (a, ω),
which has weight q rd − 1 and vanishes nowhere on Ωr . The functions g and ∆ in
Example 4 merely constitute a very special instance of this construction. We further let,
for u ∈ (n−1 )r ,
eu (ω) := eω (u1 z1 + · · · + ur zr ),
the n -division point of type u of φω . If u 6∈ Ar , eu (ω) vanishes nowhere on Ωr ,
and it can be shown that in this case,
(4.6.1)
1
e−
u = E1,u .
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Part II. Section 4. Drinfeld modules and local fields of positive characteristic
251
We are interested in the behavior around the boundary of M Γ of these forms. Let us
first describe the set {F V } of boundary divisors, i.e., of irreducible components, all of
codimension one, of M Γ − MΓ . For Γ = Γ(1) = GL(r, A), there is a natural bijection
e Pic(A)
{F V } →
(4.6.2)
described in detail in [ G1, VI 5.1 ]. It is induced from V 7→ inverse of Λr−1 (V ∩ Ar ).
(Recall that V is a K -subspace of dimension r − 1 of K r , thus V ∩ Ar a projective
module of rank r − 1 , whose (r − 1) -th exterior power Λr−1 (V ∩ Ar ) determines
an element of Pic(A). ) We denote the component corresponding to the class (a) of
an ideal a by F (a) . Similarly, the boundary divisors of M Γ for Γ = Γ(n) could be
described via generalized class groups. We simply use (4.5.1) and (4.5.2), which now
give
(4.6.3)
e Γ(n) \ GL(r, K)/P1 (K).
{F V } →
We denote the class of ν ∈ GL(r, K) by [ν]. For the description of the behavior of
our modular forms along the F V , we need the partial zeta functions of A and K . For
more about these, see [ W ] and [ G1, III ].
Definition 12. We let
ζK (s) =
X
|a|−s =
P (q −s )
(1 −
− q 1−s )
q −s )(1
be the zeta function of K with numerator polynomial P (X) ∈ Z[X]. Here the sum is
taken over the positive divisors a of K (i.e., of the curve C with function field K ).
Extending the sum only over divisors with support in Spec(A), we get
X
ζA (s) =
|a|−s = ζK (s)(1 − q −d∞ s ),
06=a⊂A ideal
where d∞ = degFq (∞). For a class c ∈ Pic(A) we put
X
ζc (s) =
|a|−s .
a∈c
If finally n ⊂ K is a fractional A -ideal and t ∈ K , we define
X
ζt mod n (s) =
|a|−s .
a∈K
a≡t mod n
Among the obvious distribution relations [ G1, III sect.1 ] between these, we only mention
(4.6.4)
ζ(n−1 ) (s) =
|n|s
ζ0mod n (s).
q−1
We are now in a position to state the following theorems, which may be proved following
the method of [ G1, VI ].
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252
E.-U. Gekeler
Theorem 4. Let a ∈ A be non-constant and c a class in Pic(A). The modular form
∆a for GL(r, A) has vanishing order
ordc (∆a ) = −(|a|r − 1)ζc (1 − r)
at the boundary component F c corresponding to c .
Theorem 5. Fix an ideal n of A and u ∈ K r − Ar such that u · n ⊂ Ar , and let
1
e−
u = E1,u be the modular form for Γ(n) determined by these data. The vanishing order
ord[ν ] of E1,u (ω) at the component corresponding to ν ∈ GL(r, K) (see (4.6.2) ) is
given as follows: let π1 : K r → K be the projection to the first coordinate and let a
be the fractional ideal π1 (Ar · ν). Write further u · ν = (v1 , . . . , vr ). Then
ord[ν ] E1,u (ω) =
|n|r−1
(ζv mod a (1 − r) − ζ0mod a (1 − r)).
|a|r−1 1
Note that the two theorems do not depend on the full strength of properties of M Γ
as stated without proofs in the last section, but only on the normality of M Γ , which is
proved in [K] for A = Fq [T ], and whose generalization to arbitrary Drinfeld rings is
straightforward (even though technical).
References
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A. Weil, Basic Number Theory, Springer 1967.
FR 6.1 - Mathematik Universität des Saarlandes
Postfach 15 11 50 D-66041 Saarbrücken Germany
E-mail: gekeler@math.uni-sb.de
Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local fields